---
title: "Swinging Spring & Internal Resonance Interactive Lab | Sandbox Physics"
description: "Can a bounce lend its energy to a swing? Yes, when the bounce frequency is close to twice the swing frequency."
canonical_url: "https://sandboxphysics.com/labs/swinging-spring/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/swinging-spring/"
last_modified: "2026-10-27"
---

M088 · One spring, two kinds of motion

# Swinging Spring & Internal Resonance

**Can a bounce lend its energy to a swing?** Yes, when the bounce frequency is close to twice the swing frequency. Then a tiny sideways nudge grows: energy flows from stretching into swinging, then back again, over and over. The total energy stays nearly constant through the exchange.

Stretch the spring, give the bob a small nudge and let go. Tune the ratio of the two small-amplitude frequencies and watch energy pass back and forth between stretching and swinging. Add a sideways launch to see the bob move in three dimensions.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

A point mass hangs on an ideal spring that pulls only along its own length, and it can move in three dimensions. The spring has no mass, does not buckle and has no damping. The motion is simulated, not measured.

Physics tutorial

## The bounce that lends its energy to a swing

**Background** A mass hanging on a spring can bounce along the spring and swing in two sideways directions. Once the displacement is no longer tiny, gravity couples these motions.

**Why it matters** You can watch energy move from one motion to another with no motor driving it and no damping.

Start with the essentials

- **Focus question**: Why can a mostly vertical release grow into a large swing?
- **One-sentence intuition**: When the small-amplitude bounce frequency is close to twice the swing frequency, a tiny sideways seed can draw energy out of the bounce.

### Core mathematical model

#### Equation of motion in Cartesian coordinates

$$
\ddot{\mathbf r}=-k(r-\ell_0)\frac{\mathbf r}{r}-\hat{\mathbf y}
$$

Units are chosen so that the mass and gravity both equal one. The vertical coordinate points up, and the spring force always points along the line to the pivot.

#### Small-amplitude bounce and swing frequencies

$$
\omega_r=\sqrt{k},\quad\omega_s=1,\quad\ell_0=1-\frac1k
$$

The equilibrium length with the mass hanging is fixed at one. These frequencies come from the linearized equations; they are not measured from the nonlinear motion.

#### Energy of the bounce

$$
E_r=\frac{\dot r^2}{2}+\frac{k(r-1)^2}{2}
$$

The spring potential is written about the loaded equilibrium, so it already includes the part of gravity that acts along the spring.

#### Energy of the swing

$$
E_s=\frac{|\dot{\mathbf r}|^2-\dot r^2}{2}+r+y,\quad E=E_r+E_s
$$

This split of the total energy is exact and never negative. The swing part holds the sideways kinetic energy and the gravity term that depends on the spring's direction. Neither part is conserved on its own.

#### Angular momentum about the vertical axis

$$
L_y=z\dot x-x\dot z=\mathrm{constant}
$$

It stays constant because the spring force is central and gravity exerts no torque about the vertical axis.

### Common difficulties

#### A swing needs a seed

**Typical misconception** If I release it exactly vertically, it will start swinging sooner or later.

**Better mental model** A motion that starts exactly vertical stays exactly vertical, because nothing in it breaks the symmetry. Use the tilt or sideways control to give it a small sideways start.

#### Internal resonance needs no drive

**Typical misconception** When the swing grows, energy is being added to the system.

**Better mental model** The energy comes from the bounce. This model is free and conservative, with no motor and nothing that varies the stiffness in time.

### Run the experiment

1. 01

   #### Release near resonance

   Choose Resonant exchange and press Play. Scrub through the whole record.

   **What to observe:** The orange stretch bar shrinks while the blue swing bar grows. Then the energy flows back.

2. 02

   #### Detune and compare

   Choose Detune the spring. It starts with the same stretch and tilt.

   **What to observe:** The sideways motion grows much less. The loaded equilibrium length stays the same.

3. 03

   #### Remove the seed

   Choose Perfectly vertical.

   **What to observe:** The bob bounces straight up and down in exact simple harmonic motion, and the swing energy stays at zero.

4. 04

   #### Open the third direction

   Choose Spatial launch and compare the front and top views.

   **What to observe:** The trail leaves the starting plane, and the angular momentum about the vertical axis stays constant.

5. 05

   #### Check the numerical accuracy

   Turn on Use half timestep and compare the largest recorded energy error.

   **What to observe:** The second-order integrator gives a smaller energy error, and the physical starting state does not change.

Physical conclusion

The exact total energy stays nearly constant while its bounce and swing parts trade back and forth. At finite amplitude the motion drifts away from what the linear frequencies predict.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)